Random walks, fractons, and electrons on percolation structures at criticality.

We discuss the localization of random walks, vibrational excitations ("fractons"), and electronic wave functions on self similar percolation clusters at criticality. We show that, contrary to the common belief, the localization behavior of random walks differs basically from the behavior of fractons and electrons. This is seen best in the shortest path (chemical) metric: While the distribution of the probability densities at fixed chemical distance l from the origin of the random walk is very narrow (for fixed time t), the distribution of the amplitudes of fractons and electrons at fixed l from their localization center is logarithmically broad (for eigenfunctions with fixed energy E or frequency w). Here we present two different approaches that account for the different localization phenomena considered. The treatments describe satisfactorily the numerical results, in particular the appearance of different localization regimes and the logarithmic dependence of the crossovers on the number of averaged configurations. [ABSTRACT FROM AUTHOR]